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5 votes

Let $f\colon (a,b)\to \mathbb{R}$ be nondecreasing and continuous. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and } f'(x)=0\}$, then $\lambda(f(E))=0$

The following proof is a more explicit version of the one already mentioned above. If anything remains unclear, I’ll be happy to provide clarification. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an ...
Jan's user avatar
  • 432
3 votes
Accepted

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

The squeeze theorem is not needed here, just the definition of $\lim_{a \to -\infty}$. To simplify the notation, set $f(t) = V_F([t, b])$ and $L = V_F(-\infty,b]$. In the quoted proof it has been ...
Martin R's user avatar
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1 vote

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

I would rewrite the proof as follows: Let $\epsilon$ be a positive number. Choose a finite increasing sequence $\{t_i\}_{i=0}^n$ of numbers that belong to $(-\infty,b]$ and satisfy $$ \sum_{i=1}^n|F(...
Beerus's user avatar
  • 2,483
1 vote

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

For each $m\in\mathbb{N}$, let $\epsilon_m=\frac{1}{m}$ and choose an increasing finite sequence $\{t_i^m\}_{i=0}^{n_m}$ of numbers that belong to $(-\infty,b]$ and satisfy $$ \sum_{i=1}^{n_m}|F(t_i)-...
Grand Minister's user avatar
1 vote

Surreal numbers by Knuth, the "bad numbers" proof.

The key here is that $x\leq z$ means, by definition, that there does not exist $z_R$ or $x_L$ with certain properties. So if you have a statement like "$x\le y$ and $y\le z$ and $x\le z$", ...
Mark S.'s user avatar
  • 24.3k
1 vote
Accepted

Applying the linearity of $f$ in the proof of Proposition 1.5 in Brezis

What you have shown in your work is correct. The choice of $t_{0} := \frac{f(x_{1})-\alpha }{f(x_{1})-f(x_{0}) }$ does not imply that $f(x_{t_{0}}) = \alpha$ and there are counterexamples under the ...
Dean Miller's user avatar
  • 2,086

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